Irreversible dynamics on Poisson manifolds

Abstract

We present a geometric construction of irreversible dynamics on Poisson manifolds that satisfies the axioms of metriplectic mechanics and the GENERIC framework. Our approach relies solely on the underlying Poisson structure and its deformation theory, without requiring any additional metric structure. Specifically, we show that if the second Lichnerowicz-Poisson cohomology group of a Poisson manifold is nontrivial, one can construct a symmetric bracket that generates irreversible dynamics compatible with energy conservation and entropy production. This bracket is derived from a 2-cocycle that deforms the original Poisson structure, thereby modifying the associated Casimir foliation. We illustrate the construction with two finite-dimensional examples and one infinite-dimensional example: the duals of the Lie algebras of the special Euclidean group SE(2), the Galilei group SGal(3) and the group of diffeomorphisms over the circle. These examples demonstrate the applicability of the method in classical mechanics, control theory, and mathematical physics.